Question

The integral domain of which cardinality is not possible:
A. 5
B. 6
C. 7
D. 10

• A. A and B only
• B. A and C only
• C. B and D only
• D. C and D only

Hint:

A crucial fact to memorize for abstract algebra questions: The order of a finite field is always a prime power (\(p^n\)). Since every finite integral domain is a field, its order must also be a prime power. This immediately rules out any integer that is not of the form \(p^n\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question relates to the structure of finite rings. An integral domain is a commutative ring with a multiplicative identity (unity) and no zero-divisors (if \(ab=0\), then \(a=0\) or \(b=0\)). A key theorem in abstract algebra connects finite integral domains to fields.

Step 2: Key Formula or Approach:
The fundamental theorems we need are: 1. Every finite integral domain is a field. 2. The order (or cardinality) of any finite field must be a power of a prime number, i.e., \( p^n \) for some prime \( p \) and integer \( n \ge 1 \).
Therefore, the cardinality of a finite integral domain must be a prime power. We need to check which of the given numbers are not prime powers.

Step 3: Detailed Explanation:
Let's analyze the given cardinalities:
A. 5: 5 is a prime number, so it can be written as \( 5^1 \). This is a prime power. Thus, an integral domain of cardinality 5 is possible (it would be the field \( \mathbb{Z}_5 \)).
B. 6: 6 can be factored as \( 2 \times 3 \). This is not a power of a single prime. Therefore, no integral domain (or field) of cardinality 6 can exist.
C. 7: 7 is a prime number, so it can be written as \( 7^1 \). This is a prime power. An integral domain of cardinality 7 is possible (the field \( \mathbb{Z}_7 \)).
D. 10: 10 can be factored as \( 2 \times 5 \). This is not a power of a single prime. Therefore, no integral domain of cardinality 10 can exist. The question asks for which cardinalities are not possible. Based on our analysis, these are 6 and 10.

Step 4: Final Answer:
The cardinalities that are not possible for an integral domain are 6 (B) and 10 (D).